If the sum of first m terms of an AP is the same as the sum of its fir

1.	If the sum of first m terms of an AP is the same as the sum of its first n terms, show that the sum of its first (m+n) terms is zero.
Answer» S_m = S_n⇒ m/2 [2a+(m-1)d] = n/2 [2a +(n -1)d] ⇒ 2a(m -n) + d(m² -m -n² + n) = 0 ⇒ (m -n) [2a +(m+n -1) d] = 0 or S_m+n = 0

If the sum of first m terms of an AP is the same as the sum of its first n terms, show that the sum of its first (m+n) terms is zero.

Answer»

S_m = S_n⇒ m/2 [2a+(m-1)d] = n/2 [2a +(n -1)d]

⇒ 2a(m -n) + d(m² -m -n² + n) = 0

⇒ (m -n) [2a +(m+n -1) d] = 0 or S_m+n = 0

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If the sum of first m terms of an AP is the same as the sum of its first n terms, show that the sum of its first (m+n) terms is zero.

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